Isothermal Parker wind ($w-\ln w=4\ln x+4/x-3$)

A steady spherical Parker wind with constant isothermal sound speed $c$ and negligible wind self-gravity obeys $4\pi r^2\rho u=\dot{M}$ and $(u-c^2/u)u^{\prime }=2c^2/r-GM/r^2$. A regular sonic point has $u=c$ at $r_s=GM/(2c^2)$. With $x=r/r_s$ and $w=u^2/c^2$, integration gives the displayed equation for the smooth accelerating transonic branch. Its slope at the sonic point is $u^{\prime }(r_s)=c/r_s$. Maintaining constant gas temperature during expansion generally requires heat input, so this model is distinct from an adiabatic wind with $\gamma >1$.

On the escaping supersonic isothermal Parker wind branch, $w=u^2/c^2\to \infty$. Its integrated equation gives $w=4\ln x+\ln w-3+4/x$, so $w/(4\ln x)\to 1$ and the title result follows. A further iteration yields $w=4\ln x+\ln (4\ln x)-3+o(1)$. The leading logarithmic growth holds for any supersonic branch with a fixed finite integration constant, but does not describe a subsonic isothermal breeze. The unbounded speed is a formal consequence of indefinite isothermal heating in a prescribed central gravitational potential.